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Det \{Γ\Η\Γ) - Eòrr>\ = 0, (45) that is, (l\H\l)-E (2|#|1) (l\H\2) (2|#|2)-£ (l|ff|S)... ( 2 | # | 3 ) . . | = 0. (46) Let E = Er be a solution of (46). If (Er — Ef) is not too large one may obtain approximate solutions using perturbation theory. One must allow for the fact that the central field levels are usually degenerate; the central field energies depend only on the configuration and for a given configuration there will be a number of linearly independent 1. COMPLEX ATOMS 29 functions Ψ(Γ).

1. an antisymmetric two-electron wave function may be written as the product of a spin function and a function of τντ2\ W(n1l1n2l2SLMsML\xvx2) = φ(5Μ5\σνσ2)φ±(η111η212ΣΜ^τντ2). The symmetric function φ+(τντ2) the antisymmetric function φ^(τντ2) inequivalent electrons we have φ±(η111η212ΣΜ^τντ2) (113) goes with the singlet spin state, and with the triplet spin state. For two = ψ:{φ{ηι11η212ΙΜ^τ1,τ2) ]/2 ± φ(η111η212ΣΜι\τ2,τ1)} (114) with φ(η111η2Ι^Μ^τ1,τ2) = — Ρ η Λ (^) — Pn%h(r2) x φ(ΙχΙ^Μ L\r νϊ 2) (115) and ^AIML|rlfra) = Σ ft^^^^^irjy^ir,).

For the individual spin functions we have, from (65 a and b), SMs il ΜΗΑΛΓδ^,σ,) ò^aMWoè 10 (ΙΙ^2)[δ(ί\σι)δα\σ2) + WKWÏIa,)] IT ίίϊΚΜϋ*,) 00 (1/V"2j[- <5(Ìh) - YMYM + Yu(^)Yii(h)l The functions φ(ΙΙΣΜΙ]τντ2) are seen to be symmetric for L = 2, anti­ symmetric for L = 1, and symmetric for L = 0.